Signatures of Branched Coverings

TL;DR

This paper investigates the signatures of branched coverings over the Riemann sphere with finite order local monodromies, identifying cases where the monodromy group is solvable and related to algebraic functions expressible in radicals.

Contribution

It classifies signatures of elliptic and parabolic types for branched coverings, determining their monodromy groups and solvability, and links these to solvable differential equations and algebraic functions.

Findings

Monodromy groups are solvable for most signatures studied.

Related differential equations are solvable in quadratures or algebraic functions.

Inverse Chebyshev polynomials can be expressed in radicals.

Abstract

In this paper we deal with branched coverings over the complement to finitely many exceptional points on the Riemann sphere having the property that the local monodromy around each of the branching points is of finite order. To such a covering we assign its \textit{signature}, i.e. the set of its exceptional and branching points together with the orders of local monodromy operators around the branching points. What can be said about the monodromy group of a branched covering if its signature is known? It seems at first that the answer is nothing or next to nothing. Indeed, generically it is so. However there is a (small) list of signatures of \textit{elliptic} and \textit{parabolic} types, for which the monodromy group can be described completely, or at least determined up to an abelian factor. This appendix is devoted to investigation of these signatures. For all these signatures (with one exception) the corresponding monodromy groups turn out to be solvable. Linear differential equations of Fuchs type related to these signatures are solvable in quadratures (in the case of elliptic signatures --- in algebraic functions). A well-known example of this type is provided by Euler differential equations, which can be reduced to linear differential equations with constant coefficients. The algebraic functions related to all (but one) of these signatures are expressible in radicals. A simple example of this kind is provided by the possibility to express the inverse of a Chebyshev polynomial in radicals. Another example of this kind is provided by functions related to division theorems for the argument of elliptic functions. Such functions play a central role in the work [1] of Ritt.